What do you mean by "acceptable"? "Recorded in the Bible as well"? or "not contradictory to what the Bible says"?
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AlypiusMar 8 '13 at 16:48

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This question will be greatly improved with a simple statement in the question asking about the scope of works you care about. Narnian is exactly correct if you are asking about Canonical sources (i.e. books most Christians accept) or non-Canonical ones. Early in the Christian era, there were all sorts of books written that contain all sorts of stories. They aren't "Gospel" or even Biblical - but they are old.
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Affable GeekMar 8 '13 at 18:04

3 Answers
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There was a manuscript that began circulating in the 2nd century written by someone who called himself "Thomas the Israelite," claiming that Jesus was able to do miracles as a child. The manuscript was later given the title, "Infancy Gospel of Thomas", and you can see a full English translation here.

The first story in this manuscript tells of Jesus' creation of sparrows from clay.

This child Jesus, when five years old, was playing in the ford of a mountain stream; and He collected the flowing waters into pools, and made them clear immediately, and by a word alone He made them obey Him. And having made some soft clay, He fashioned out of it twelve sparrows. And it was the Sabbath when He did these things. And there were also many other children playing with Him. And a certain Jew, seeing what Jesus was doing, playing on the Sabbath, went off immediately, and said to his father Joseph: Behold, thy son is at the stream, and has taken clay, and made of it twelve birds, and has profaned the Sabbath. And Joseph, coming to the place and seeing, cried out to Him, saying: Wherefore doest thou on the Sabbath what it is not lawful to do? And Jesus clapped His hands, and cried out to the sparrows, and said to them: Off you go! And the sparrows flew, and went off crying.

Then will Allah say: "O Jesus the son of Mary! Recount My favour to thee and to thy mother. Behold! I strengthened thee with the holy spirit, so that thou didst speak to the people in childhood and in maturity. Behold! I taught thee the Book and Wisdom, the Law and the Gospel and behold! thou makest out of clay, as it were, the figure of a bird, by My leave, and thou breathest into it and it becometh a bird by My leave, and thou healest those born blind, and the lepers, by My leave. And behold! thou bringest forth the dead by My leave. And behold! I did restrain the Children of Israel from (violence to) thee when thou didst show them the clear Signs, and the unbelievers among them said: 'This is nothing but evident magic.'"

In another story in the Infancy Gospel of Thomas, a teacher named Zacchaeus attempts to teach Jesus the Greek alphabet.

And a certain teacher, Zacchaeus by name, was standing in a certain place, and heard Jesus thus speaking to his father; and he wondered exceedingly, that, being a child, he should speak in such a way. And a few days thereafter he came to Joseph, and said to him: Thou hast a sensible child, and he has some mind. Give him to me, then, that he may learn letters; and I shall teach him along with the letters all knowledge, both how to address all the elders, and to honour them as forefathers and fathers, and how to love those of his own age. And He said to him all the letters from the Alpha even to the Omega, clearly and with great exactness. And He looked upon the teacher Zacchaeus, and said to him: Thou who art ignorant of the nature of the Alpha, how canst thou teach others the Beta? Thou hypocrite! first, if thou knowest. teach the A, and then we shall believe thee about the B.

Most early Christians denied the authenticity of this work. For example, second century apologist Irenaeus of Lyons referred to the story about the alphabet in declaring the work spurious.

Besides the above [misrepresentations], they adduce an unspeakable number of apocryphal and spurious writings, which they themselves have forged, to bewilder the minds of foolish men, and of such as are ignorant of the Scriptures of truth. Among other things, they bring forward that false and wicked story which relates that our Lord, when He was a boy learning His letters, on the teacher saying to Him, as is usual, “Pronounce Alpha,” replied [as He was bid], “Alpha.” But when, again, the teacher bade Him say, “Beta,” the Lord replied, “Do thou first tell me what Alpha is, and then I will tell thee what Beta is.” This they expound as meaning that He alone knew the Unknown, which He revealed under its type Alpha.

The Infancy Gospel of Thomas concludes with the story of Jesus in the temple at age 12, taken from the Gospel of Luke.

So to answer your question, yes there was an early book claiming Jesus did miracles at a young age, but (A) it was written more than 100 years after he lived, and (B) it was not considered authentic even at the time. It is not likely that this book contains any authentic stories about Jesus that are not found in the canonical gospels.

46 After three days they found him in the temple, sitting in the midst of the teachers, listening to them and asking them questions,
47 and all who heard him were astounded at his understanding and his answers.

This story invites admiration by merely presenting Jesus "in the role of the faithful Jewish boy, raised in the traditions of Israel, and fulfilling all that the law requires." It does not present him as a youthful miracle-worker.

Turning water into wine at the wedding in Cana is said to be the first of His signs. John 2:11:

11 Jesus did this as the beginning of his signs in Cana in Galilee and so revealed his glory, and his disciples began to believe in him.

This seems to establish this sign as the first public sign.

The focus of the Bible seems to be on establishing Jesus as a normal person of good reputation in the time before his ministry. If any stories of Jesus as a youthful miracle-worker were known, surely they would have been known by those in his hometown of Nazareth. But instead, they react with surprise in Mark 6:1-6:

2 When the sabbath came he began to teach in the synagogue, and many who heard him were astonished. They said, “Where did this man get all this? What kind of wisdom has been given him? What mighty deeds are wrought by his hands!

So the answer is, no, such childhood miracles do not seem to be in accord with the Bible.

The work that you are thinking of is the Gospel of St Thomas, a work that is rejected and typically seen as a primarily pagan/Gnostic composition. It presents the Child Jesus as a typical "child-god", performing miracles such as bringing toy birds made of clay to life (as mentioned, this story seems to have made it into the Quran), being disobedient towards his teachers and parents, developing a bad reputation for both himself and his father, and generally behaving uncharitably towards others. Consider this story, in which the fig-tree of later years is replaced by a young child, who seemingly did a wrong on par with kicking over a sandcastle:

Behold, even now you shall be dried up like a tree, and you shall not bring forth either leaves, or root, or fruit. And straightway that boy was quite dried up.

And the boy died. Again, these stories are non-canonical and rejected as heretical. The surviving versions seem to have had the most offensive elements removed. The CE section on the Gospel of St. Thomas:

There are two Greek and two Latin redactions of it, differing much from one another. A Syriac translation is also found. A Gospel of Thomas was known to many Fathers. The earliest to mention it is St. Hippolytus (155-235), who informs us that it was in use among the Naasenes, a sect of Syrian Gnostics, and cites a sentence which does not appear in our extant text. Origen relegates it to the heretical writings. St. Cyril of Jerusalem says it was employed by the Manichæans; Eusebius rejects it as heretical and spurious. It is clear that the original Pseudo-Thomas was of heterodox origin, and that it dates from the second century; the citations of Hippolytus establish that it was palpably Gnostic in tenor. But in the extant Thomas Gospel there is no formal or manifest Gnosticism. The prototype was evidently expurgated by a Catholic hand, who, however, did not succeed in eradicating all traces of its original taint. The apocryphon in all its present forms extravagantly magnifies the Divine aspect of the boy Jesus. In bold contrast to the Infancy narrative of St. Luke, where the Divinity is almost effaced, the author makes the Child a miracle-worker and intellectual prodigy, and in harmony with Docetism, leaves scarcely more than the appearance of humanity in Him. This pseudo-Gospel is unique among the apocrypha, inasmuch as it describes a part of the hidden life of Our Lord between the ages of five and twelve. But there is much that is fantastic and offensive in the pictures of the exploits of the boy Jesus. His youthful miracles are worked at times out of mere childish fancy, as when He formed clay pigeons, and at a clap of His hands they flew away as living birds; sometimes, from beneficence; but again from a kind of harsh retribution.